Math Problem Statement
Solve the application by writing a system of equations that models the conditions and then applying inverse matrix methods.
The following table shows the carbohydrate, fat, and protein content of three food types. Food Type Carbohydrate (g) Fat (g) Protein (g) I 13 10 13 II 4 4 3 III 1 0 10 A nutritionist must prepare two diets from these three food groups. The first diet must contain 46 grams of carbohydrate, 34 grams of fat, and 72 grams of protein. How many grams of each food type are required for the first diet? Food Type I
Incorrect: Your answer is incorrect. g Food Type II
Incorrect: Your answer is incorrect. g Food Type III
Incorrect: Your answer is incorrect. g
The second diet must contain 22 grams of carbohydrate, 14 grams of fat, and 66 grams of protein. How many grams of each food type are required for the second diet? Food Type I
Incorrect: Your answer is incorrect. g Food Type II
Incorrect: Your answer is incorrect. g Food Type III
Incorrect: Your answer is incorrect. g
Solution
To solve this problem using the inverse matrix method, we set up a system of equations for each diet based on the food content table and the nutritional requirements.
Step 1: Define the variables
Let:
- = grams of Food Type I
- = grams of Food Type II
- = grams of Food Type III
The system of equations can be expressed in matrix form as:
13 & 4 & 1 \\ 10 & 4 & 0 \\ 13 & 3 & 10 \end{bmatrix} \begin{bmatrix} x_1 \\ x_2 \\ x_3 \end{bmatrix} = \begin{bmatrix} 46 \\ 34 \\ 72 \end{bmatrix}$$ for the first diet, and $$\begin{bmatrix} 13 & 4 & 1 \\ 10 & 4 & 0 \\ 13 & 3 & 10 \end{bmatrix} \begin{bmatrix} x_1 \\ x_2 \\ x_3 \end{bmatrix} = \begin{bmatrix} 22 \\ 14 \\ 66 \end{bmatrix}$$ for the second diet. ### Step 2: Matrix Representation Let: - $$A = \begin{bmatrix} 13 & 4 & 1 \\ 10 & 4 & 0 \\ 13 & 3 & 10 \end{bmatrix}$$ - $$\mathbf{x} = \begin{bmatrix} x_1 \\ x_2 \\ x_3 \end{bmatrix}$$ - $$\mathbf{b_1} = \begin{bmatrix} 46 \\ 34 \\ 72 \end{bmatrix}$$ (for diet 1) - $$\mathbf{b_2} = \begin{bmatrix} 22 \\ 14 \\ 66 \end{bmatrix}$$ (for diet 2) The solution is given by: $$\mathbf{x} = A^{-1} \mathbf{b}$$ ### Step 3: Compute $$A^{-1}$$ and Solve We compute the inverse of $$A$$, then multiply it with $$\mathbf{b_1}$$ and $$\mathbf{b_2}$$ to find $$\mathbf{x}$$. I will calculate this now.import numpy as np # Define the matrix A and vectors b1, b2 A = np.array([[13, 4, 1], [10, 4, 0], [13, 3, 10]]) b1 = np.array([46, 34, 72]) # First diet requirements b2 = np.array([22, 14, 66]) # Second diet requirements # Compute the inverse of A A_inv = np.linalg.inv(A) # Solve for x1 (first diet) and x2 (second diet) x1 = np.dot(A_inv, b1) x2 = np.dot(A_inv, b2) x1, x2(array([3., 1., 3.]), array([1., 1., 5.]))(array([3., 1., 3.]), array([1., 1., 5.]))successThe solutions for the required grams of each food type are: ### First Diet: - Food Type I: $$3 \, \text{g}$$ - Food Type II: $$1 \, \text{g}$$ - Food Type III: $$3 \, \text{g}$$ ### Second Diet: - Food Type I: $$1 \, \text{g}$$ - Food Type II: $$1 \, \text{g}$$ - Food Type III: $$5 \, \text{g}$$ These values satisfy the nutritional requirements for both diets. Let me know if you would like a detailed breakdown of the calculations or have any additional questions. --- ### Related Questions: 1. How can the inverse matrix method be applied to systems with more equations or variables? 2. What happens if the matrix $$A$$ is singular and cannot be inverted? 3. How do we verify the correctness of the solution in such matrix-based problems? 4. What are the limitations of the inverse matrix method compared to other methods for solving linear systems? 5. Can we use software like Python or MATLAB to automate similar matrix-based calculations? ### Tip: Always verify that the determinant of the coefficient matrix is non-zero before attempting to compute its inverse. A zero determinant indicates that the system has no unique solution.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Linear Algebra
Matrix Operations
Systems of Linear Equations
Formulas
Matrix equation: A * x = b
Inverse matrix formula: A^-1
Theorems
Matrix Inversion Theorem
Properties of Matrix Multiplication
Suitable Grade Level
Grades 10-12 or Undergraduate Level
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